Viscosity, entropy and the viscosity to entropy density ratio; how perfect is a nucleonic fluid?
Aram Z. Mekjian Department of Physics and Astronomy, Rutgers University, Piscataway, NJ 08502, TRIUMF Laboratory, 4004 Westbrook Ave. , Vancouver, B.C. CA V6T 2A3
Abstract The viscosity of hadronic matter is studied using a classical evaluation of the scattering angle and a quantum mechanical discussion based on phase shifts from a potential. Semi classical limits of the quantum theory are presented. A hard sphere and an attractive square well potential step are each considered as well as the combined effects of both. The lowest classical value of the viscosity for an attractive potential is shown to be a hard sphere limit. The high wave number-short wavelength limits of the quantum result have scaling laws associated with it for both the viscosity and entropy. These scaling laws are similar to the Fraunhoher diffraction increase for the hard sphere geometric cross section. Specific examples for nuclear collisions are given. The importance of the nuclear tensor force and hard core is mentioned. The viscosityη , entropy density s andη / s ratio are calculated for a gas of dilute neutrons in the unitary limit of large scattering length. Away from the unitary limit, the ratio of the interaction radius or the scattering length to the interparticle spacing introduces a variable y besides the fugacity z . The isothermal compressibility is shown to impose important constraints. The results forη / s are compared to the AdS/CFT string theory minimum of (1/ 4π )h / kB to see how close a nucleonic gas is to being a perfect fluid. Theη / s ~1 h / kB for a neutron gas in its unitary limit. Theη / s ~ 3 h / kB treating the nuclear scattering as billiard ball collisions. The minimumη / s for a neutron gas occurs in regions of negative isothermal compressibility and high fugacity where higher virial terms are important. In a neutron-proton system higher virial terms are associated with a liquid-gas phase transition and critical opalescent phenomena. A connection between the nuclear flow tensor and viscosity is developed using a Fokker-Planck equation and a relaxation time description. The type of flow- laminar, vortex, turbulent- is investigated.
PACS 24.10.Pa, 21.65.Mn, 66.20.-d
I. Introduction
Understanding the viscosity of matter is of interest in many regimes of energy and in several different areas of physics. These areas include atomic systems, nuclear matter, neutron star physics, low energy to relativistic energy heavy ion collisions and at the extreme end string theory. For example viscosity plays a role in collective flow phenomena in medium and relativistic heavy ion collisions (RHIC). Specifically, viscosity inhibits or resists flow in such collisions. Ref. [1,2] are early theoretical studies of viscosity and flow. Some recent experimental results for RHIC physics can be found in Ref. [3] and some overviews are in Ref. [4,5]. A microscopic kinetic theory description
1 of viscosity involves the transport of momentum across an area and involves particle interactions through mean free path arguments. If interactions are strong, the shear viscosity is small. The shear viscosity is inversely proportional to the scattering cross section. Substances with low kinematic viscosity and with high Reynolds number do not flow smoothly as in laminar flow, but rather form eddies when flowing around obstacles. The Reynolds number also depends on the flow velocity and the eddy behavior is easily seen by putting a paddle in water in a boat moving at various speeds. Surprisingly, in relativistic heavy ion collisions, the quarks and gluons act as a strongly coupled liquid [6,7,8] with low viscosity rather than a nearly ideal gas of asymptotically free particles with high viscosity. Low viscosity fluids which interact strongly are called nearly perfect fluids. String theory has put a small lower limit on the ratio of shear viscosityη over entropy density s given byη / s ≥ h /(4πkB ) [9] with kB the Boltzmann constant. The string theory result has generated considerable interest in questions concerning strongly correlated systems and viscosity. The focus of the present paper is on low and moderate energy systems of interacting nucleons. A system of nucleons has parallel properties such as correlated behavior, bound states, phase transitions and critical point behavior. Nearly perfect fluids with very low viscosity appear also in cold atoms. Properties of interacting quantum degenerate Fermi gases were first observed in atomic systems [10- 12]. Atoms are cooled in a laser trap to the point where quantum statistics and an associated thermal wavelength play an important role. Such systems can be tuned by using a magnetic field and properties of Feshbach resonances are used to study the strong coupling crossover from a Bose –Einstein condensate of bound pairs to a Bardeen- Cooper-Scrieffer BCS superfluid state of Cooper pairs. A remarkable aspect of strongly interacting Fermi gases is a universal behavior which occurs when the scattering length is very large compared to the interparticle spacing. In this unitary limit, properties of a heated gas are determined by the density ρ and temperatureT , independent of the details of the two body interaction. Early theoretical discussions of dilute Fermi systems at infinite scattering length can be found in Ref. [13,14]. At temperatureT = 0 , the Fermi energy E of a strongly interacting Fermi gas differs from the Fermi energy EF of a non- interacting Fermi gas by a universal factorξ with E = ξEF . Accounting for this difference in nuclear systems is referred to as the Bertsch challenge problem [15]. Initial work on this problem was done by Barker [16] and latter Heiselberg [17]. A Monte Carlo numerical study of the unitary limit of pure neutron matter is given in Ref. [18]. Analytic studies of pure neutron systems can be found in the extensive work of Bulgac and collaborators [19-21]. In these studies the dimensionless factorξ ≈ 0. 4 . Part of present paper is an extension of an earlier work [1] in which the viscosity of hadronic matter was studied using a relaxation time approximation to the Boltzmann equation and also a Fokker-Planck description. This earlier paper focused on features associated with collective flow. Specifically, the kinetic flow tensor was related to the pressure tensor and the collective velocity field. The pressure tensor, in turn, was related to the nuclear viscosity. The Kubo-Green formulae [22,23,24] relates the viscosity to time fluctuations in the pressure tensor. A calculation of the Reynolds number showed that the flow is laminar. A Fokker Planck approach has been recently used [25] to study the viscosity of the quark-gluon phase. A relaxation time approach also appears in Ref. [26,27] for trapped Fermi gas in a oscillator well near the unitary limit of large scattering
2 length. Questions related to entropy in nuclear systems were also studied early on [28] using the Sakur-Tetrode law and the Saha-equation. Recently, the behavior of entropy in the unitary limit was given [29]. The work presented in this paper gives a more refined calculation of the viscosity than given in Ref. [1] as well as further discussions of the entropy. Both classical and a quantum approaches to the viscosity are discussed using a Chapman-Enskog description [30,31]. The classical calculation involves the scattering angle while the quantum approach relates the viscosity to properties of the scattering phase shifts. In such studies a potential must be specified. The interactions used here to describe the nuclear scattering are a square well potential, a pure hard core and a combination of the two. The pure hard core potential pictures the collisions as arising from impenetrable billard balls and is used as a comparison. The discussion of viscosity from a pure hard core potential also contains new results which show a scaling law regime with increasing momentum of the colliding pair of nucleons. This scaling feature parallels results that arise when considering scattering from a hard sphere [32] which lead to a diffractive increase by a factor of two from the classical result ofπR 2 . The square well potential with a hard core approximates a nuclear potential and the role of the nuclear hard core as well as the nuclear tensor force are mentioned . A study of viscosity using a delta-shell potential can be found in Ref.[33,34]. The focus here will be on a pure one component system- such as in a gas of neutrons. One important feature of the interaction between nucleons is the very large scattering length asl and its associated unitary regime. The unitary limit of the viscosity and entropy is examined for this system. A study of Feshbach resonances and the second virial coefficient in atomic systems is given in ref. [35,36]. Viscosity to entropy considerations also appear in the damping of giant resonances in nuclear physics [37] and in atoms in laser traps [38].
II. Classical and quantum approaches to the viscosity
II. A. Mean free path and relaxation time approaches
First, the standard discussion of the viscosity that can be found in textbooks such as Ref. [39] relate the viscosity to the number density ρ , mass of a fluid particle m , mean ) speed v = 8k BT / mπ of a Boltzmann distribution and mean free pathlλ as
1 η = ρmv)l . (1) 3 λ
2 The mean free pathlλ = 1/(ρσ ) . The scattering cross sectionσ = πD for hard sphere scattering of particles with diameter D . For this description the viscosityη no longer depends on the number density ρ and is simply
1 1 η = mv) . (2) 3 πD2
) Theη does not involve Planck’s constant. If the mean free path is taken aslλ = vτ , with
3 τ the mean time between collisions, theη ~ ετ . Theε = E /V ~ mv) 2ρ is the energy density and E is the energy per particle. The ρ now explicitly appears inη but is removed as follows. Taking the entropy density as s ~ ρkB , the ratioη / s ~ Eτ / kB is again ρ independent. The product Eτ is governed by an uncertainty relation [2,4,5] in a quantum approach, giving η / s ~ h / kB with the particular constant of proportionality1/ 4π from string theory. In a relaxation time approximation to the Boltzmann equation [1], the collision term is replaced with(df / dt) = −( f − f 0 ) /τ with f 0 the local equilibrium coll R phase space distribution. The viscosity is thenη = ρτ R kBT and is directly related to a ) 2 relaxation timeτ R . The mv ~ kBT which leads to η ~ ρEτ R as discussed above.
II.B. Chapman-Enskog Theory
The Chapman-Enskog theory[30,31] relates the viscosityη to terms involving the scattering angle χ or phase shiftδ l . Fig. 1 shows scattering off various potentials.
FIG.1. Scattering off various potentials. Left figure is hard sphere scattering. Middle figure is scattering off an attractive potential. Snell’s law of reflection applies to the hard sphere case and gives the scattering angle χ or angle of deflection with respect to the initial direction as χ = π − 2θi . Snell’s law of refractionsinθi = nsinθ f applies to the attractive well and has
χ = 2(θ f −θi ) . Right figure is the combined effect of an attractive square well with an inner hard core. The right figure is discussed in section II.B.5.
Angular momentum conservation leads to Snell law of refraction nsinθ f = sinθi , withsinθi = b / R , and
2µV0 2µ V0 2 n = n(E,V0 ) = (E + V0 ) / E = 1+ 2 2 = 1+ 2 λ . (3) h k h
4 Theb is the impact parameter and E is the incident energy. This equivalence with an index of refraction shows why the reflection and refractionn(E,V0 ) of light can be explained in terms of both Huygens wave picture and classical Newtonian mechanics. Fermat’s principle that the path taken is the one of least time leads to Snell’s equations for reflection and refraction. The particle/quantum picture is due to Feynman and represents the process as a sum of all phasors over all possible paths that a photon will take in the case of light. The Fresnel equations determine the reflected and refracted intensities by boundary conditions on the E&M waves at an interface. Corresponding, the phase shifts for particles are obtained in a similar manner from the wave function. The wavelength dependence given in Eq. 3 arises from the deBroglie relationλ = h / p . The viscosity is obtained from [30,31]
5 πk Tm η = B (4) 8 πR2ω(2,2)
with ω (2,2) given by
1 2 (2) ω (2,2) ≡ ω = dγ ⋅ e−γ γ 7φ (E,V , R) (5) πR 2 ∫ 12
(2) where φ12 (E,V ,R) ≡ φ is evaluated both classically and in a quantum approach. The 2 2 2 γ = E / kBT with E = h k / 2µ and with µ the reduced mass. The classical approach uses
∞ φ = 2π (sin 2 χ)b ⋅ db . (6) ∫0
The quantum calculation evaluates φ using
4π (l +1)(l + 2) φ = sin 2 (δ − δ ) . (7) 2 ∑ l+2 l k l 2l + 3
The δ l is the l'th phase shift of the potential used to describe the scattering. For identical particles, the factor 4π is replaced with 8π and the sum is over evenl for bosons and odd l for “spinless” fermions. Spin effects for fermions can also be included – see appendix A. The total cross section for scattering is simply
4π 2 σ = 2 ∑ (2l +1)sin δ l (8) k l and this expression has some features that parallel the expression forφ . The cross section appears in Eq. 1 for the viscosity through the factor involving the mean free path. A somewhat related quantity that will appear in expressions for the entropy [29] is the Beth-
5 Ulhenbeck continuum integral [40-42] which is labeled BC and given by :
1 dδ B = (2l +1) l exp(− 2k 2 / mk T )dk . (9) C ∑ ∫ h B π l dk
II. B.1 Classical calculation of viscosity for a hard sphere potential.
For hard sphere of radius RC scattering, the impact parameter b = RC cos χ / 2 and 2 2 (2,2) bdb = −(1/ 4)RC (sin χ)dχ . Thus φ = 2πRC / 3and thereforeω = 2 . The viscosity is
5 πkBTm η = 2 . (10) 16 πRC
This expression for the viscosity can be compared to that given by Eq. 1 with 2 ) σ = πRC and v = 8kBT / mπ leading to
1 ) 1 8 πkBTm η = ρmvlλ = 2 2 . (11) 3 3 π πRC
The ratio of these two expressions forη is (5/16)/ (8 / 3π )=1.04, a difference of 4%. 2 2 The differential cross section is σ (χ) = −(b / sin χ) ⋅ db / dχ = RC / 4 and total σ = πRC which is the geometric cross section since anything hitting the sphere is scattered.
II.B.2 Classical calculation of viscosity for a square well potential.
For a square well θ f = θi + χ / 2, sinφ f = sinθi / n and thus
b b2 b2 b b2 b2 sin χ = 2 1− (1− 2 ) − 2 1− (1− 2 ) . (12) nR n2R2 R2 R R2 n2R2
The connection between b and χ can also be written as
χ χ sin n sin b = 2 = 2 . (13) R 1 2 χ χ 1+ − cos 1+ n 2 − 2n cos n 2 n 2 2
When the index of refraction n → 1, χ → 0 and when n → ∞ ,0θ f → , and χ → −2θi .
6 The differential scattering cross section can be obtained fromσ (χ) = b / sin χ) db / dχ . Letting z = cos(χ / 2) , σ (χ) =)n2R2 (nz −1)(n − z) /(4z(1+ n2 − 2nz)2 . Theσ (χ) is constrained by nz −1 ≥ 0 . The largest scattering angle χ is whenb = R , sinθi = 1and sinθ f = 1/ n . The z = cos(χ / 2) = cos(θr − π / 2) = sinθr =1/ n . At this point nz −1 vanishes andσ (χ) = 0 . Theσ = πR2 , which is the sameσ as that of a hard sphere. The 2 viscosity based on Eq. 1 would then have lλ = 1/ ρπR . However, the Chapman-Enskog (2) approach requires an evaluation of φ12 (E,V0,R) ≡ φ and thenω to obtainη . Theφ is
φ / 2πR2 = (16 − 30n − 40n2 + 20n3 + 40n4 + 4n5 + 20n7 − 30n9 ) /120n4 +
(15(n2 −1)(n8 −1) /120n4 )log[(n +1) /(n −1 )]. (14)
Two limits can be considered. One limit is for n near unity and the other for large n . For n near unity, n = 1+ ε withε small. Then φ = 2πR2 (− ε 2{2logε + 2log2 + 3} − ε 3 /3) and thus φ,ω → 0 if n → 1andη → ∞ . However in this limit of infinite viscosity, the concept of momentum transport from collisions between layers of fluid fails since the particles move back and forth between the endpoints defined by the moving walls of the container. In subsection II.B.5 a hard core will be include inside the attractive potential and φ,ω will not longer go to zero. For large n ,φ = 2πR2 (1/3 − 2/(35n) −1/(3n2 )...) . The (2) 1/3 term in the parenthesis gives the hard sphere result. The φ12 (E,V0,R) can be 2 substituted into the integral forω , using n = 1+ hT / γ with hT ≡ V0 / kBT , which in turn determinesη . The general behavior of ω with hT = V0 / kBT is shown in Fig. 2. ω / 2 1.0 0.9 0.8 0.7 0.6 0.5 0.4
5 10 15 20 25 hT
FIG. 2: (Color online) The classical behavior ofω / 2 versus hT = V0 / kBT for an attractive potential. The limit hT → ∞ hasω / 2 → 1 which is the hard sphere limit. The rise ofω/2 to the value1is approximately exponential with 0.8 ω ≈ 2 (1− exp(−0.25(hT ) )). The exact calculation shown in the figure slightly overshoots 1 which will be neglected. The exponential representation has a
7 slightly higher value at low hT .
The results of Fig. 2 show that the classical calculation of the viscosity over a broad range of hT = V0 / kBT from an attractive potential can be approximated as
5 πkBTm η = 2 0.8 . (15) 16 πR (1− exp(−0.25(V0 / kBT ) ))
Therefore, in a classical evaluation of the viscosity, the smallest value ofη for an attractive interaction is at the largest value ofω which is the hard sphere result.
II.B.3 Quantum calculation of viscosity for a hard sphere potential and the semi-classical limit h → 0 .
First, results for hard sphere scattering will be given and compared to the classical evaluation. The phase shifts for a hard sphere are simplytanδl = jl (x) /ηl (x ) with jl and
ηl Bessel functions. The x = kRC , with RC the hard sphere radius and k the wave number. The quantitiesφ,ω,η , as well as the entropy which will be used later, are all developed in appendix A. Theω can be rewritten as
∞ 4 −ξx 2 7 1 (l +1)(l + 2) 2 ω = 4ξ dx ⋅ e x sin (δl +2 (x) − δl (x)) (16) ∫0 2 ∑ x l =0,1,2,... 2l + 3
2 withξ = (λT / RC 2π ) . The quantum wavelengthλT = h / 2πkBTm is a de Broglie 2 wavelength associated with the thermal momentum E ~ kBT ~ p / 2m . Theγ that appears inω isγ = (x / 2π )(λT / RC ) . The φ sum has the following scaling property 2 when x → ∞ : φ → 2πRC / 3which is the classical hard sphere scattering result mentioned above. The scaling behavior of φ(x ) is shown in Fig.3. This scaling result parallels a 2 similar result for the cross section which in the high energy limitσ → 2πRC . This factor 2 of two increase over the hard sphere geometrical areaπRC arises from diffraction.
) φ (x )
1.15
1.10
20 40 60 80 100 120 140
x
8
) FIG. 3. (Color online) Scaling property of φ (x) with x.The quantity plotted is ) 2 2 φ ≡ (6 / x )Σl [(l +1)(l + 2) /(2l + 3)]⋅ sin (δ l+2 (x) − δ l (x))versus x . For this rescaled 2 ) quantity the limiting value is unity. Theφ = 4πRCφ / 6.
The other extreme energy is a low energy result. At very low energies only an S − wave 2 phase shift is important. The cross section goes to4πRC , or four times the geometrical 2 result. Also, φ → 8πRC / 3for x << 1. The S − wave phase shift isδ 0 = −kRC = −x , giving 2/3 for the bracket term in the integration for small x since sin2 x ≈ x2 . The integration of exp(−ξx2 )x7 is 3/ξ 4 resulting inω = 8 compared to the classical valueω = 2 . Corrections from spin and identical particles can be included and are given in appendix A. For now, these corrections will be neglected since a comparison is made with the classical calculation which does not contain these factors. The range ofη is then
5 πkBTm 5 πkBTm 2 ≥ η ≥ 2 . (17) 16 πRC 64 πRC
2 The behavior of ω is determined by ξ = (λT / RC 2π ) which in turn depends on the temperatureT through λT . LowT has low associated energies and large λT . When 2 λT / RC >> 1, only small x contribute to the integral because exp(−ξx ) suppresses large x . Both extreme endpoints in Eq. (17) do not involveh . The semi-classical limit has h → 0 andξ → 0 . In this limit, the scaling behavior shown in Fig. 3 arises andη is the classical value. To see how the viscosity evolves from the S − wave limit to the classical value, small x expansions are made for the P, D, F phase shifts as given in appendix A.
These results forδ 0 ,δ1,δ 2 ,δ3 ,... can be used to obtain an expansion forω for small x , further expanded in evenl and oddl componentsωE andωO , as
32 1 360 256 48 1152 ω = 8 S − S + 3 ( S,D + D ).. + 2 P − 3 P + .. ≡ ωE + ωO . (18) 3ξ ξ 7 35 E ξ 5ξ O
The factor8 is the pure S − wave result. The 1/ ξ 2 term arises solely from a P − wave. The contribution of each partial wave is also given. It should be noted that the result of 2 2 2 Eq. (18) gives an expansion forω in inverse powers of h since ξ = (λT / RC 2π ) ~ h . The viscosity is connected to this series expansion around the S − wave scattering limit using Eq. (4). The hard sphere quantum result forη is shown in Fig.4 as a function of ξ .
η
ηCl
9 0.42 0.40 0.38 0.36 0.34 0.32
5 10 15 20 25 30 ξ → FIG. 4: (Color online) The hard sphere quantum viscosity as a function of ξ .
The vertical axis isη /ηCl , the ratio of the quantum result for the viscosity divided by the classical hard sphere result which is given by Eq. (10) with ω = 2 . At low values ofξ , the quantum calculation is the same as the classical result. At largeξ the quantum result is ¼ the classical result.
II.B.4 Quantum calculation of viscosity for a square well potential and the unitary limit
Square well phase shifts are determined by the boundary conditions of continuity of the wave function and its slope at the square well radius. The phase shiftδl is given by the equation tanδ l = {kjl′(x) − γ l jl (x)}/{knl′(x) − γ l nl (x)} withγ l = αjl′(y) / jl (y ) . The 2 2 α = k + V0 2µ / h , y = αR , x = kR . The prime superscript on the spherical Bessel functions represent derivatives with respect to x or y . The y = n(E,V0 )x withn(E,V0 )
= 1+ V0 / E the index of refraction of the classical description. Again, the low energy behavior will be considered as a baseline for comparison. In this limit the S − D wave 2 2 phase shift δ2 − δ0 ≈ −δ0 and the bracket term in Eq. (16) is 2sin δ0 /3x . The S − wave
δ 0 = arctan[(kR /αR) tanαR − kR . An effective range approximation forδ 0 reads 2 k cotδ0 = −1/ asl + r0k / 2 . The scattering lengthasl = R(1− tanα 0 R /α 0 R )and the 2 3 2 2 effective range is r0 = R −1/α0 asl − R /3asl . Theα0 = 2µV0 / h . For large asl , r0 ≈ R .
A zero energy bound state appears whenα 0 R = π / 2 . Then asl → ∞ . Similarly, for a zero energy resonant like state asl → −∞ . In an effective range approximation the ω
2 ∞ 2 1 (ka ) ω = 4ξ 4 dx ⋅ e−ξx x7 sl (19) ∫0 2 2 x 2 2 2 r0 1+ asl (asl − r0 )k + (asl (asl − r0 )k ) 2 4(asl − r0 )
10 As a further approximation, r0 terms can be neglected when asl >> r0 . Then
2 2 + ζ (ζ −1) − eζ ζ 3Γ(0,ζ ) ω = 4 ξ (20) 3 2
2 2 2 whereζ = ξR / asl = (λT / asl 2π ) and Γ(0, g) is a gamma function, with the special ∞ −t −1 case Γ(0, g) = E1(g) = e t dt . The limit asl → ∞ is referred to as the unitary limit. In ∫g 2 the unitary limitω → 8ξ /3 = 8(λT / R 2π ) /3.Thusω introduces quantum effects via the factor λT . The S − wave unitary or universal thermodynamic limit forη is determined by the quantum wavelengthλT and is independent of the radius of the potential:
πk Tm 3/ 2 15 B 15 (πkBTm) 15 h η → 2 = 2 = 2π 3 . (21) 32 λT 16 h 32 λT
The last equality in Eq. (21) shows that the viscosity is proportional to Planck’s constant 3 divided by the quantum volume λT and also related to a thermal momentum p ~ kBTm 3 2 −23 3 as η ~ p / h . At kBT = 10MeV , the result of Eq.(21) is 1.03⋅10 MeV ⋅ s / fm and at −23 3 kBT = 10MeV , it is η = 0.032 ⋅10 MeV ⋅ s / fm . By contrast the S − wave hard sphere 2 limit can be written asη = (5 2π /16) ⋅ h /(λT πRC ) but is h independent. For identical particles, the 4π is changed to8π in φ and the sum is over even or odd l states. For particles with spin, spin factors appear. Identical spin 1/ 2 fermions interacting through a S − wave,l = 0 state are coupled to a total spin zero singlet state. This introduces an additional factor of ¼ inφ . The net effect is to reduceφ by ½, thereby increasingη for fermions to twice the value given above to:
15 h η = 2π 3 . (22) 16 λT
The unitary limit forη is independent of the potential used since it is based on an effective range result and with asl → ∞ . A calculation of η with a delta shell potential [33] gave the same result and also the same result can be found in Ref. [26,27].
II.B.5 The role of the nuclear tensor force and hard core
Some remarks on the role of the nuclear tensor force are as follows. The quantum 2 theory of viscosity hasη ~ Σ(l +1)(l + 2) /(2l + 3) ⋅ sin (δ l+2 − δ l ) . The nuclear non-central 3 3 tensor force couples the S1 phase shift to the D1 . In general, channels with spin S = 1and angular momentum L = J −1 are coupled to channels with spin S = 1and angular
11 momentum L = J +1. The analysis of phase shifts involves a parameter labeledε J [43]. The centrifugal barrier suppresses the wave function at low energies where the phase 2l+1 2l shift δ l ~ k and at these energiesε J ~ k . In the isospin I = 0 channel ( np system) the 3 nucleon-nucleon S1 starts at the valueπ because of the deuteron bound state and decrease rather quickly, reaching a value of ~π / 2 within the laboratory energy range of 3 5MeV . (See Fig.2-34, P264 in Ref. [43]) .Thereafter the S1 phase shift decreases only 3 slightly over the energy range 5MeV − 50MeV . Note that the low region nearδ[ S1 ] ~ π 2 3 does not contribute toη since sin π = 0 . The D1 phase shift starts at a value equal to 0 and becomes somewhat significant at energies above 15MeV . Thus features related to the tensor force are suppressed for low enough temperatures. In the isospin I =1 channel, 1 the nn, pp, np S − wave interaction is in the spin singlet S = 0 state or S0 state. The tensor force does not act on a singlet state. In the I =1channel, the nn,pp, np P − wave 3 interaction is in the spin triplet S = 1state which has J = 0,1, 2 components or P0,1,2 states. The tensor force couples a spin triplet P − wave to a spin tripletl = 3 F − wave 3 3 whereJ = 2,3, 4 . Since J must be the same, the tensor coupling is between P2 and F2 . Again, the tensor force couples only becomes important at higher energies because the centrifugal barrier suppresses thel = 3wave function compared to thel = 1wave function. Calculations of the effect of a hard core can be incorporated into a potential model. An analysis was represented in Ref. [29] for S − wave phase shifts arising from a square well interaction with a hard core. The S − wave phase shift for this potential is given b 2 2 2 δ 0 = arctan[(kR0 /αR) tanα(R0 − RC ) − k(R0 − RC ) − kRC . Theα = k + α 0 and 2 α0 = 2µV0 / h . In an effective range approximation, the scattering length is asl = R0 (1− tanα0 (R0 − RC ) /α0R) and the effective range is now r0+C = r0 + r0C with 2 2 2 2 r0C = RC (1− 2R0 / asl + R0 / asl +1/α0 asl ) . The hard core renormalizes asl and r0 . The RC is the radius of the core and R0 is the radius of the attractive square well. Results for higher partial waves where also given in Ref. [29]. The modifications due to a hard core in a classical approach are shown in the right figure in Fig. 1. Depending on the impact parameterb , index of refractionn(E,V0 ) and radius of the hard core RC the particle either misses the hard core or intersects it. For a given index of refraction the division between the two trajectories occurs at an impact parameterbm determined by nsinθ f = nRC / R0 = sinθi = bm / R0 or simplybm = nRC . For b > bm , theχ = −2(θi −θ f ) and the results of Eq. (12) can be used. If b ≤ bm , the χ is determined byχ = 2θβ − π − 2(θi −θ f ) , sinθβ / R0 = sin(π −θ f ) / RC . When RC → R0 ,
θ β → π −θ f and χ = π − 2θi , which is the hard sphere χ . Subsection II.B.2 hasφ,ω → 0 , andη → ∞ as n →1. However with a hard core and with n ≈ 1, the scattering is basically off the hard core sincebm = nRC ≈ RC and the results of subsection II.B.1 apply. For large n such thatbm = nRC ≥ R0 then all impact parameters lead to trajectories that intersect the hard core. For n → ∞ all trajectories are directed toward the center inside the
12 attractive region. Such trajectories then get reflected backwards against the incident path and leave the attractive region at the incident angle in the forward direction so that
χ = π − 2θi . The result is again reflection off a hard sphere but now off a sphere of 2 2 radius R0 . The ratio of theφ integrals at the limits n → 1and n → ∞ is then RC / R0 .
II.B.6 Low energy behavior of the viscosity of a dilute neutron gas
The viscosity of a dilute gas of neutrons will now be considered. A previous study of the second virial coefficient showed that the S − wave approximation accurately described the scattering up to temperatures of ~15 MeV before P − wave and D − wave contributions start to become significant. In a space symmetric l = 0 S − wave state, the neutrons are coupled to a total spin S = 0 antisymmetric state. In this channel the observed S − wave scattering length is asl = −17.4 fm and the effective range is r0 = 2.4 fm. A potential of the type considered in this paper that reproduces these properties has an attractive depth of 31.4MeV with a radius of 2 fm and hard core of radius 0.27 fm [29]. Theω integral given by Eq. (20), when corrected for an effective range contribution, leads to a viscosity
( a + r ) 2 15 η = sl 0 π h (23) 3 ζ 3 asl 2 + ζ (ζ −1) − ζ e Γ(0,ζ )16 λT
2 2 withζ = λT /(2πasl (asl − r0 )) = (hc) /(mc(asl (asl − r0 )kBT) = 0.12 /(kBT /1MeV ) . The factor(asl − r0 ) / asl = 19.8 /17.4 = 1. 138. The last bracket term in Eq. 23 is the unitary limit. The middle square bracket term is equal to 0.9 at kBT = 0.5MeV , where ζ = .24 and is equal to 0.99 at kBT = 10MeV ,whereζ = 0. 012 . Thus the viscosity of a neutron gas is within ~10% of its unitary limit when kBT ≥ 0.5MeV . In the above equation the factor in square bracket is very accurately approximated by1+ ζ /3for the entire range ofζ . The ζ / 3 is obtained from the asymptotic value of2 +ζ (ζ −1) −ζ 3eζ Γ(0,ζ ) for largeζ . The value1is the unitary limit. Thus the viscosity is very accurately given by
( a + r ) λ2 15 η = sl 0 + T 2π h . (24) 2 3 asl 3⋅ 2πasl 16 λT
The first term alone in the first bracket is larger than the unitary limit since 2 ( asl + r0 ) / asl >1. The second term, with an overall 1/(asl λT ) dependence, becomes large when λT /( 2π asl ) ~ 3 . For asl = 17.4 fm , the λT ~ 76 fm and thusT ~ 1/20MeV for the second term to become comparable to the first term.
II. Entropy and viscosity to entropy density ratio
13 III.A.1 General considerations
The equation of state EOS can be used to obtain the interaction part of the entropy using the Maxwell relation (∂P / ∂T)V = (∂S / ∂V )T . The EOS to second order in the virial ˆ 2 2 expansion isP = kBT (A /V − b2 A /V ) . The energy including the interaction energy [29] obtained from the thermodynamic identity(∂E / ∂V )T = T (∂P / ∂V )T − P leads to 2 ) E(V ,T) = (3/ 2)AkBT + kBT (db2 / dT) . The temperature dependence of E(V,T) and
S(V ,T) are connected by (∂E / ∂T)V = T(∂S / ∂T)V . Also (∂E / ∂V )T = T (∂S / ∂V )T − P. ) ) ) ) Theb2 = b2,exc + b2,int where b2,exc is the exchange part of the virial coefficient and ) ) 7 / 2 3 b2,int arises from interactions. Theb2,exc = ± (1/ 2 )λT / gS with g S = 2S +1and the + sign is ) for bosons and the minus sign is for fermions. The interactionb2,int is
) b E (J ) 1 dδ 2,int = g exp(− b ) + g J exp(−bk 2 )dk ≡ B + B . (25) 3 3/ 2 2 ∑ J ∑ J ∫ b C λT 2 / g S Eb ( J ) kBT π J dk
2 ) Theb = λT / 2π . The Bb is the bound part of b2,int withEb (J )the energy of a bound state with spin J and degeneracy g J . The second term involving dδ l / dk is a term due to Beth and Uhlenbeck [40-42] and it reflects continuum correlations and is labeled BC . The ) 2 ) interaction part of the entropy is obtained from b2,int using Sint = kB (A /V )d(Tb2,int ) / dT . 2 2 2 The interaction entropy can also be written in terms ofξ = b / RC = λT / 2πRC as a variable which is useful when evaluating Sint for a hard core potential. Specifically,
2 3/ 2 A 2 2 3/ 2 2 d 1/ 2 Sint = −kB 2 (2πRC ) ξ ξ BC (ξ ) . (26) V gS dξ
A similar expression applies in an effective range approximation withξ replaced with 2 2 ζ == λT /(2π (asl (asl − rO )) and RC replaced byasl (asl − rO ) .
III.A.2 S-wave effective range results and the unitary limit
The S − wave effective range approximation leads to [29]
dδ0 asl r0asl 2 = − 2 2 4 (1+ k ) ≈ dk 1+ asl (asl − r0 )k + (r0asl ) k / 4 2
asl r0asl 2 − 2 (1+ k ) . (27) 1+ asl (asl − r0 )k 2
14 The last approximation in Eq. (27) omits the k 4 term which turns out to accurately 4 describe the behavior of BC unless asl ~ r0 . Also if r0 > aSl > 0the k is necessary to keep the integral from diverging. Away from these regions, the BC is given by
a (2a 2 − 3r a ) b b a 2 r B − sl sl 0 sl exp( )Erfc( ) − sl 0 . (28) C 2 3/ 2 2 2 2 4(asl − asl r0 ) asl − asl r0 (asl − asl r0 ) 4(asl − asl r0 ) πb
) The unitary limit is asl → ±∞ . In this limit, the continuum contribution BC tobint is
1 1 r0 1 r0 3 1 2 r0 3 2π r0 3 BC = − sign[asl ]⋅ − + ( ) = − + ( ) . (29) 2 4 π b 32 π b 2 4 λT 16 λT
When asl → +∞ , the bound state term of 1 and continuum term − sign[asl ]/ 2 = −1/ 2 give +½ which is the same result for a zero energy resonance with g J =1. The r0 / λT part of BC does not contribute to the Sint . The Sint in the limit asl → −∞ is
A2 23/ 2 1 3 2π r 3 S = −k λ3 − 0 . (30) int B T 2 3 V gS 4 16 λT
Even in the limit of infinite scattering length, features related to the effective range persist [29]. For Sint the residual effects of the effective range have a cubic dependence.
III.A.3 Entropy for a neutron gas
For spin ½ fermions, and in particular for a pair of neutrons, the S − wave interaction is in the spin singlet S = 0 state while the P − wave interaction is in the spin triplet S = 1state 3 3 3 and hasJ = 0,1, 2 . The J − weighted average of the P0 , P1, P2 phase shifts given by 3 3 3 3 δ ( P) = {5δ ( P2 ) + 3δ ( P1) +1δ ( P0 )}/9 is small [43]. Both a nuclear spin-orbit force and tensor force are necessary to explain features associated with the P − wave phase shifts. r r A spin-orbit force given byVLS (r)L ⋅ S = VLS (r)(J (J +1) − l(l +1) − s(s +1)) / 2 cannot account for the behavior of the triplet P − wave phase shifts. If the spin-orbit were the 3 only spin dependent force the P1 J = 1phase shift would be intermediate between the 3 3 r r P2 , J = 2 and P0 , J = 0phase shifts from the J (J +1 ) dependence in L ⋅ S . This feature is not consistent with experimental results shown in Fig. 5. The D − wave interaction is in 1 1 the spin singletS = 0, J = 2 state and hasδ ( D2 ) . Theδ ( D2 ) is small as shown in Fig. 5. Thus the neutron gas entropy is mainly dominated by the S − wave term. In an effective range theory the S − wave entropy is given by Eq. (30). .
15
FIG. 5. The nucleon-nucleon isospin T=1 phase shifts in radians versusELab (MeV ). The figure appears in part in Bohr and Mottleson [43].
III.A.4 Entropy of a hard sphere gas
For a hard sphere gas the bound state contributionBb = 0 and BC is obtained from 2 2 2 tanδl = jl (x) /ηl (x) , x = kRC and dδl / dx = −1/(x ( jl (x) +ηl (x))) . Appendix A gives a 2 2 2 complete discussion of Sint and alsoη . Usingξ = b / RC = λT / 2πRC , the S − wave −1/ 2 contribution toBC = −ξ / 2 and the interaction entropySint = 0 by Eq. (26). At very high energy, or as x → ∞ , the following scaling relation [29] was noted:
x 1 2 S ≡ (2l 1) x4 . (31) 2 ∑ + 2 2 → l=0 ( jl (x) +ηl (x)) 3
In the limit that many terms contribute to BC
1 ∞ dδ 2 R 3 B = dk (2l +1) l exp(−bk 2 ) = − π C . (32) C ∫ ∑ 3 π l=0 dk 3 λT
For spin zero bosons and spin ½ fermions these scaling or geometric limits are reduced by ½ -see appendix A. For high T the interaction entropy is then
1 4 A2 S → − ( πR3 )k . (33) int 2 3 C B V
2 3 The value Sint = −VCkB A / 2V , with hard core volumeVC = 4πRC / 3, is the semi classical limit of the interaction entropy. The result can be obtained from the quantum result by takingh → 0 orξ → 0 . The Sint is density A/V dependent but temperatureT independent. This result will be compared with a van der Waals gas in the next subsection. For a monatomic ideal gas of A nucleons, the entropy is given by the Sakur-Tetrode 5/ 2 3 expression [28] which isS = Sid = AkB ln(e VgS / AλT ).
16
5 Vg 1 Aλ3 Aλ3 A d ) S = Ak + ln S + {± T + 2(2−4 − 3−5/ 2 )( T )2} + (Tb ) … (34) B 3 7 / 2 2,int 2 AλT 2 VgS g SV V dT with the + sign for fermions and the – sign for bosons in the ± signs in S .
II.A.5. Comparison with a van der Waals gas; comments on the role of a liquid/gas phase transition, critical point fluctuations and critical opalescence.
2 2 The EOS of a van der Waal gas is(P + a(T )(A /V ))(V − bex A) = AkBT . The bex is the excluded volume term and the a(T) arises from two particle attractive interactions which can be temperature dependent. Using the Maxwell identity listed above, the entropy is 3/ 2 2 S = AkB ln(V − bex A)T + (A /V )(da(T) / dT) + CS , withCS a constant. Spin entropy will be omitted. The Sakur-Tetrode law for the entropy is a quantum theory result for an 3 3/ 2 ideal gas (no b, a(T) ) and givesCS = − AkB ln(Ah /(2πmk) ) + (5 / 2)AkB . A constant
(5/ 2)AkB also appears in the Sakur-Tetrode law from E + PV . The entropy is then
2 5/ 2 (V − bex A) da(T ) A S = AkB ln e 3 + . (35) AλT dT V
In the dilute limit theV >> bex A. Expanding the logarithm gives
2 5/ 2 V A da(T) A S ≈ AkB ln e 3 − kB A bex + . (36) λT ^ V dT V
2 The − kB A bex /V factor is the same as the scaling limit of the hard sphere quantum gas apart from a factor of 2 reduction for identical particles. One of the interesting features associated with the van der Waals EOS is its connection with a liquid gas phase transition when combined with a Maxwell construction which is introduced to eliminate regions of negative isothermal compressibility. A review of the nuclear liquid gas phase transition can be found in Ref. [44]. The van der Waal model parallels a density functional approach based on a Skyrme interaction [45,46]. The behavior of viscosity with temperature in a liquid is considerable different than in a gas. The viscosity of a liquid decreases rather rapidly withT while that of a gas increases with T [30,31]. As an example of the rapid decrease with T is the viscosity of water which decreases by a factor of about 6 from its freezing point 2730K to its boiling point3730K . The result of Eq. (1,2) gives a slow increase of η as T for a gas from v) , the mean speed of the particles. The ratio for η(3730K) /η(2730K ) = 373/ 273 = 1.17 . In the unitary limit, η ~ T 3/ 2 andη(3730K) /η(2730K ) =1.173 =1.6. A second interesting feature is the presence of a critical point where large fluctuations in density occur. The phenomena of critical opalescence is a characteristic feature of
17 these critical point density fluctuations where the scattering cross section increases dramatically because droplets of all sizes are present. A simple model is the Fisher model [47] of a critical point where the distribution of cluster sizes falls as a scale τ invariant power law. The number of clusters nk of size k varies as nk ~1/ k withτ a critical exponent. The present work evaluates the viscosity of one type of particle with a single fixed size. The viscosity of a given type of cluster depends on mass m and radius R = D / 2 of a cluster as m / D2 ~ 1/ R1/ 2 using the simple expression of Eq. 2. The radius term comes from the cross section through the mean free path and mass term is present 3 from speed factors. The mass of a cluster with A nucleons varies as m ~ mp A ~ R . At a critical point, the matter is now made of many different types of clusters, but subject to a constraint of overall mass conservation which reads A = Σk knk . Because of clustering, the multiplicity m = Σk nk is reduced from A to a much smaller number. Scattering can now also occur between particles of different sizes. In a ratioη / s the entropy must be considered when bound states are formed. The entropy is greatly affected by large changes in the multiplicity. To see this feature consider the factor (5 / 2)kB that appears in the entropy. Each particle (monomer, dimer,…) adds 3/2 kbT to the energy and kBT to PV . SinceTS = E + PV − Σµk nk k for a mixture of non-interacting ideal gases, the entropy S = 5.mk T / 2 − µ n B ∑k k k A second way to explore the behavior of a system around a critical behavior is based on an order parameter expansion. The simplest order parameter theory is a Ginzburg-Landau mean field approach which has been used to study the fluctuations in density near the critical point in nuclear systems [48]. Large density fluctuations lead to a divergence of the isothermal compressibility with an associated exponent describing the divergence. Similarly, fluctuations in the energy determine the heat capacity. The viscosity is determined by fluctuations in the stress tensor. Each divergence has a critical exponent. The critical exponents of mean field theories are not those observed experimentally and improved techniques based on renormalization group methods have been developed [41]. The divergence in the viscosity arises from the coupling of the transverse velocity to the order parameter density fluctuations. Past calculations [49] show that the viscosity diverges with an exponent 8/15π 2 and somewhat larger values have been noted from dynamic renormalization group techniques [50].
III.A.6. Isothermal compressibility of a dilute neutron gas in the unitary limit .
As already noted, the behavior of the isothermal compressibilityκT around a scale free critical point is an important thermodynamic quantity in such studies. The behavior ofκT in the unitary limit also has interesting features. In the dilute gas limit these features can be obtained from the second virial coefficient through the equation
1 ∂V 1 κ κ = − = = = T .ideal . (37) T ˆ 2 2 ˆ V ∂P T AkBT /V − 2b2 A kBT /V 1− 2b2 A/V
18 TheκT .ideal = 1/((A/V )kBT ) is the ideal gas compressibility. TheκT should be positive for the mechanical stability of a gas. For an ideal Bose gas the isothermal compressibility becomes infinite at the condensation point with the singularity arising from the sum of an infinite series of terms in a virial expansion. For an ideal gas these terms arise solely from symmetrization terms of the type shown in{} brackets in Eq. (34). For an imperfect Bose ˆ gas this divergence is removed [41]. For fermions, if b2 is positive, interaction effects are more important than fermionic antisymmetrization effects and κT will have a peak as T increases from low to high temperatures as discussed in Ref. [51]. The presence of a hard core potential or strong repulsive three body terms is also important in understanding the compressibility of nuclear matter. Repulsive components are necessary for saturation of cold nuclear matter at the proper density. ) 3 9/ 2 3/ 2 2 The unitary limit for a neutron gas isb2 = λT (−1/ 2 + 2 /(2 ⋅ 2 )) when effective 9 / 2 range corrections are neglected. ThenBC = 1/ 2 and the additional factor −1/ 2 is from 2 2 antisymmetrization of the pair of neutrons while the1/ 2 = 1/ g S . The compressibility in the dilute neutron gas in the unitary limit is: 1 1 κ = ξ κ , ξ = = . (38) T κ T ,ideal κ 7 A 7 1− 2( ) λ3 1− 2( )z 29/ 2 V T 29/ 2
When effective range corrections are neglected the isothermal compressibility takes on this very simple form. Effective range corrections to the compressibility can be included using Eq. (27). Since these corrections involve BC they contain both linear and cubic terms in the ratio r0 / λT and reference to interaction potential appears inκT . A neutron gas is mechanically stable whenκT > 0 and this condition is realized for low z . A similar 7 / 2 result for bosons has1/ ξκ = 1− 2 ⋅ (17 / 2 )z . For a hard sphere S − wave Fermi gas 7 / 2 2 theκT = κT ,ideal /(1+ z / 2 + (2z / gS )(2RC /λT )) is positive.
III.B Viscosity to entropy density ratio η / s
Low viscosity to entropy density is associated with a nearly perfect fluid [4,5]. How perfect is a gas of nucleons? How close is a nuclear system to the AdS/CFT string theory minimumη / s = (1/ 4π )h / kB [9], where s = S /V ? As a first step in trying to answer these questions, a simple one component system of spin ½ fermions will be considered and with little additional effort results for spin 0 bosons will be given. This study is done in a unitary limit in the next subsection B.1 neglecting a small correction to the entropy density in the unitary limit from the effective range. In subsection B.2 and B.3, theη / s ratio is studied in a system not at or close to the unitary limit of infinite scattering length. In particular the effective range theory and also the hard sphere gas are used since simple analytic results can be obtained. The main difference with the unitary limit is that a dimensionless variable (labeled y ) involving either the scattering length or the radius of the hard core to the interparticle spacing appears inη / s . Theη / s becomes a function
19 3 of y and the fugacity z = ρλT .
III.B.1 η / s in fermionic and bosonic systems in the S − wave unitary limit asl → −∞ .
As a first example only S − wave interactions in the unitary limit will be taken. The spin ½ fermion case is approximately realized in pure neutron matter where the neutron pairs are coupled to total spin S = 0 and have a large negative scattering length. The fermionic case is compared to the bosonic case for spin zero bosons. Two main differences arise between the fermionic case and the bosonic case. The first difference is the change to symmetrization for bosons from antisymmetrization for fermions. The second main difference arises from spin degeneracy factor gS and associated spin entropy. The entropy density to lowest order in antisymmetrization corrections for fermions is
5 1 23/ 2 1 s = ρk + ln g − ln ρλ3 ± ρλ3 − ρλ3 . (39) B S T 7 / 2 T 2 T 2 2 gS (g S ) 2 ⋅ 2
2 3 Theη = gS 15 2πh /(λT 64)for S − wave fermions or bosons in the unitary limit. Defining
3 µ / k BT the standard thermodynamic variable called the fugacity as z = ρλT = e , the η / s is
η 2 15 h 1 1 = gS 2π 3 / 2 (40) s j 64 kB z 5 gS 1 2 1 + ln gS − ln z + ((−1) 7 / 2 − 2 )z 2 2 gS gS 4
The minimum ofη / s or maximum of s /η occurs at zm given by
3/ 2 gS 1 2 1 3/ 2 + ln gS − ln z + 2((−1) 7 / 2 − 2 )z = 0 (41) 2 g S gS 4
which has a solutionzm, f = 3. 52 for fermions andzm,b = 0. 965for bosons. For fermions the valuezm, f = 3. 52 is in a region of z where higher order terms must be included. The problem of a large value of zm for fermions resides more in the higher order interaction effects from three body, four, …body terms rather than higher order antisymmetrization effects. Theη / s at zm, f =3.52 is
η 1 = .806 h = 10.14 h (42) s f kB 4π kB
For bosons zm,b = 0.965 and
20
η 1 = .610 h = 7.67 h (43) s b kB 4π kB
Thezm, f = 3. 52 and zm,b = 0.965 are in a regions of negative isothermal compressibility where the system is mechanically unstable. Higher order effects beyond the second virial coefficient should be included for a proper description at these high values of the fugacity. The next two subsections look at the question ofη / s away from the unitary limit.
B.2 η,s,& η / s away from the unitary limit; effective range approximation
The behavior of viscosity, entropy density and their ratio away from the unitary limit 4 asl → −∞ is developed in this subsection. Again, the k correction in the denominator of
Eq. [27] will be neglected which imposes restrictions on the relation of asl to r0 as already discussed. The regionas / > 0 contains at least one bound state and scattering off bound states must be included as well as corrections to the entropy from k 4 corrections.
Therefore, the caseas / < 0will only be developed for now. To keep the results simple the effective range r0 corrections will also be omitted. Only S − wave terms will be considered in this section which requires kBT ≤ 25MeV or λT ≥ 3 fm for a hadronic system. The viscosity for a fermionic of bosonic system is then given by
λ2 15 η = 1+ T g 2 2π h . (44) 2 S 3 3⋅ 2πasl 64 λT
For the same effective range approximation the interaction entropy is
3/ 2 3/ 2 3 A 2 (2π ) 2 asl Sint = −kB A 2 f S (ζ ) (45) V g S 4
2 1/ 2 ζ 1/ 2 ζ with fS (ζ ) = ζ [1/(2ζ )e erfc( ζ ) + ζ e erfc( ζ ) −1/ π ] . In the unitary limit 3/ 2 3 2 3 / 2 asl → −∞ ,ζ → 0 , f S (ζ ) → ζ / 2 = ( λ T /( 2 π a sl ) / 2 and the expression of
Eq. (27) results. When ζ >> 1then f S (ζ ) → 1/(2 π ) and Sint depends on the interaction 3 potential through the factor asl . The Sint is added to the ideal gas entropy to give 3 7 / 2 3 S = kB A[5 / 2 − ln(AλT /V ) − (1/ 2 )AλT /V ] + Sint . Theη / s then follows and is
F (z, y) η h 2 15 2 η = g s π (46) gS −7 / 2 s kB 64 z(5 / 2 − ln z + (−1) 2 z / g S − FS (z, y))
21 2/ 3 2 3/ 2 2 3 2/ 3 2 with Fη (z, y) = 1+ z /(3y ) andFS (z, y) = (2 / g S ) ⋅ (2y / 4) f S (z / y ) . The 2 / 3 2 2 / 3 2 3 2 f S (z / y ) = f S (ζ ) withζ = z / y = λT /(2πasl ) . A dimensionless variable y given by
1/3 y = 2π asl /(V / A) (47)
3 is introduced besides the fugacity variable z = (A/V )λT which is a thermodynamic variable. The y variable is a measure of the ratio of the scattering length to interparticle spacing. Including effective range corrections adds another variable which can be defined 1/ 3 in a manner similar to y with asl replaced by r0 ; namely x = 2π ro /(V / A) which can be viewed as a measure of diluteness since r0 ≈ R0 the radius of the well ≈ diameter of the particle for a short range nuclear potential. A plot ofη / s versus z for various y for fermions is shown in Fig. 6. Again, the minima occur in a region of negative isothermal compressibility and large fugacity.
η / s
h / kB
1.1
1.0
0.9
3 4 5 6
z
FIG. 6. (Color online) The ratioη / s in units of h / kB versus z for various y for fermions. The values of y are y =,∞ 100,25,15. The lowest curve has the lowest y and higher curves have increasing y . Taking a minimum of η / s ~ 0.75 h / kB , thenη / s ~ 9.4( h /(4πkB ) .
III.B.2 η / s for a hard sphere gas; S − wave model
Some simple models of viscosity treat the collision between particles as a hard sphere collisions. This subsection explores properties of such a gas, and in particular the η / s ratio, in a quantum description. The problem of negative isothermal compressibility does not arise as shown in III.A.6 for fermions. Only S − wave interactions are considered in this subsection. Results for all partial waves are given in the appendix. Results are shown in Fig. 7.
.
22
η η y 2 / h y 2 / h s kB s kB 0.56 0.22 0.54 0.52 0.20 0.50 0.18 0.48 0.16 0.46
0.4 0.6 0.8 1.0 1.2 1.4 1.0 1.5 2.0 2.5 3.0 z z FIG. 7. (Color online) The ratioη / s versus fugacity z for various y for a hard 2 sphere S-wave interaction. The vertical axis is y η / s divided by h / kB . The 1/ 3 y = 2π asl /(V / A) . The left figure is the Bose gas and the right figure is the Fermi gas. The lowest curve in each figure has y = 0 and each successive curve has 0.1 unit higher in y with the uppermost curve at y = 0.4 . The minima inη / s shift to higher z with increasing y . The vertical axis does not have a large spread in values since the curves are rather flat expect for the y = 0.4 curve. A comparison of the S − wave hard sphere model with the geometric limit is given in appendix A.
Theω , including an extra factor of 2 for identical fermions or bosons and a factor 2 1/ gS =¼ for spin ½ fermions coupled to spin 0, is
2 4 2 1 1 ω = 2 4ξ 11/ 2 ( ξ (9ξ − 2) + {4 + 5ξ(3ξ − 4)}⋅ D( )) (48) gS 3 8ξ ξ where D(1/ ξ ) is the Dawson F function which has an expansion for large ξ given by −1/ 2 n n n the D(ξ) =)ξ Σn=0 (−1) 2 /(ξ (2n −1)!! . The ratioη / s is then given by the equation
η h 5 2 1 1 1 = 2π 2 1/ 3 2 / 3 2 7 / 2 . (49) s kB 8 y z ω(z / y ) (5/ 2 + ln gS − ln z ± z /(2 gS )
Theξ = z2 / 3 / y2 .
IV Viscosity and collective flow in nuclear collisions 4.1 Kinetic flow in a relaxation approach to the Boltzmann equation
23 N The kinetic flow tensor is Fij = Σk=1 (Pi (k)Pi (k) / 2m(k) where i,j are components of the r momentumP(k ) of particle k which has mass m(k) = m . The sum is over all particles. The following ansatz will be used for the phase space distribution for a system of two symmetric colliding nuclei:
1/ 2 r r r 2 r r r 2 (0) r m m(v − u(r,t)) m(v + u(r,t)) f = n(r,t) r exp(− r + exp(− r , (50) 2πθ(r,t) 2θ (r,t) 2θ (r,t)
The f (0) =)f (0) (r,vr,t is taken as the zero’th order approximation to the Boltzmann equation which has a phase space density f (r,vr,t) = f . Theur(r,t) ≡ ur = < vr > and 3θ (r,t) / 2 = m < vr − ur(r,t) 2 > / 2 where the expectation values are taken withf (0) . The r n(r,t) =)d 3v f (r,vr,t . The Boltzmann equation is (∂ / ∂t + vr ⋅ ∇ + (F / m) ⋅ ∇ ) f = ∫ r ν r (∂f / ∂t)coll . The F is a force term from an external field or Hartree-Fock field. In a (0) relaxation time approximation the collision term[∂f /∂t]coll = −( f − f ) /τ R .The correction f − f (0) = g is taken to be small and the left hand side of the Boltzmann equation is evaluated by taking f = f (0) giving
1 ∂θ m 2 5 1 1 2 (0) g = −τ R U i U − + Λij U iU j − δ ijU f . (51) θ ∂xi 2θ 2 θ 3
r r r Repeated indices are summed over. The Λij = m / 2(∂ui /∂x j + ∂u j /∂xi ) andU = v − u . t r The pressure tensor P has elements given by Pij = mn(r,t) UiU j = Pδij + Pij′ = r r ρθδ ij − (2µ / m)(Λij − (m / 3)δ ij∇ ⋅ u) with µ = ρτ Rθ and ρ is the number density. An important connection is the relation of the kinetic flow tensor to the pressure tensor. This result is F = (m / 2) n(r,t)u (r,t)u (r,t)d 3r + P d 3r ≡ F (u) + F (P) . The off diagonal ij ∫ i j ∫ ij ij ij part of Fij has a term coming fromFij (u ) which is the collective flow term andFij (P′ ) which is related to the shear viscosity through the relationship
F (P′ ) = P′d 3r = −(µ / m)Λ Ω . (52) ij ∫ ij ij V
The ΩV is the volume of the system. The minus sign inFij (P′ ) shows that the shear viscosity cancels part of the collective flow. Early calculations were presented in Ref. [1] and the results showed that the cancellation could be significant.
4.1 Viscosity and Reynolds number; laminar or turbulent flow?
The Reynolds number RY is defined as RY =)(dρmu /η where d is a characteristic length.
24 The quantity η /mρ ≡νη is the kinematic viscosity so that RY = du /νη . The connection with the string theory limit onη / s arises from the connection of the entropy density to ρ since s ~ kB ρ . Thus RY ~ dm / kBη / s . As a first approximation the simple ) ) expressionη = (1/ 3)nmvlλ can be used to give RY as RY = 3(u / v) ⋅ (d / lλ ) . In a collision the characteristic distance d is of the order of the size of the nucleus. The collective velocityu is of the order of the incident velocity in a medium energy collision. The ) 2 ) thermal speed isv/c= 8kBT / mc π ~ 1/ 20 at kBT = 1MeV and v/c ~ 1/ 6 at 2 kBT = 10MeV . The collective velocity / c ≤ incident velocity / c ~ 2Ecm / mc where Ecm is the center of mass energy per particle. However, the temperature is coupled ) to Ecm via 3 kbT /2 ≤ Ecm . Thus(u / v ) ~1 or ~½ the incident energy goes into flow and ~ ½ into thermal energy. Under these conditions the Reynolds number is governed by(d / lλ ) .
The concept of viscosity fails if lλ >> d . Taking lλ = d , the Reynolds number is then ) RY ~ (u / v) ⋅ (d / lλ ) ~ 1. High Reynolds numbers occur whend / lλ >> 1 . Ford / lλ ~ 10 , 3 3 RY ~ 10 . Turbulence sets in when RY ~10 . To get RY ~10 requires a very short mean free path or a very short relaxation time sincel =< v > τ . But a short relaxation time λ R destroys the flow, driving the system to thermal equilibrium in a collision. An analysis [1], based on a Fokker-Planck equation, of the time evolution of the momentum space v r density f P (±P0 ,t) of two colliding nuclei with initial momentum ± P0 gave
2 r r −βt −2βt r Aexp− ( P m P0e / 2mkBT(1− e )) f P (±P0 ,t) = −2βt 3/ 2 . (53) [2πmkBT(1− e )]
Each of the 2 colliding nuclei have A nucleons in the overlapping fireball region. At t = 0 r r r r v r f P (+P0 ,t) + f P (−P0 ,t) = A(δ (P − P0 ) + δ (P + P0 )) . The degradation of the centroid r −βt r momentum is ± P0e , which is just the behavior of a particle started with ± P0 and r subject to a frictional force − βP . The1/ β is a relaxation time. The variance of the momentum spreads with time as kBT(1− exp(−2βt)) . An estimate oft is the collision time of two overlapping nuclei which is the radius of the nuclei RA divided by the incident velocityvinc ortcoll ~ RA / vinc . Thus to have some persistence of the initial momentum, or collective motion, βtcoll ~ 1 ~ βR / vinc . Using this last result and equating the relaxation time1/ β with the relaxation timeτ R that appears in the expression for the viscosity. The
u vinc vinc Ecoll RY ~~. (54) < v > < v > u Eth
where Ecoll is the collective energy and Eth is the thermal energy of a particle.
25
FIG. 8. Reynolds number RY and flow. At low Reynolds number the flow past an obstacle is laminar. At higher RY vortices appear behind the obstacle. At still higher RY the vortices break off and flow with the fluid. Further discussion can be found in Feynman [52]
V. Conclusions
Viscosity plays an important role in many areas of physics as discussed in the introduction. An initial study [1] showed that in a heavy ion collision viscosity reduced the collective flow [1]. Recent interest in the nature of hydrodynamic flow in RHIC experiments arose from the possibility that strongly coupled hadronic systems can behave as a perfect fluid [3-5]. Studies of nearly perfect fluids appear in atomic physics and in particular in ultracold Lithium atoms. Such studies employ the Feshbach resonance as a tuning device where an external magnetic field controls the scattering length and investigates the unitary limit where universal thermodynamics applies. Furthermore, a lower limit on the viscosity to entropy density ratio came from string theory [9] which has motivated considerable theoretical discussion and further experimental studies in several systems. Thus, a wide range of energy scales in very different systems are manifesting similar behavior. The present paper explored questions related the viscosity, the entropy and the ratio of viscosity to entropy density to see if a nuclear system at moderate energy or temperature behaves as a perfect fluid. The large experimentally observed scattering length in nucleon-nucleon systems makes this system very useful for studies of the unitary limit and universal thermodynamic behavior at moderate temperatures of several to tens of MeV and at various densities. The viscosity was studied in both a classical and quantum approach for several types of potentials. These include treating the collisions between nucleons as: A) billiard ball hard spheres scattering which is often used as a model in textbooks to discuss viscosity; B) interactions represented by an attractive square well tuned to the experimentally determined effective range and scattering length of nucleon-nucleon collisions; C) a combination of a short range repulsion and a longer range attraction which represents features associated with realistic interactions. The classical theory of the scattering angle was cast into a form that contains Snell’s laws of reflection and refraction with an energy dependent index of refraction. The lowest classical value of the viscosity of the attractive potential is the hard sphere limit. The quantum theory involved calculation of the phase
26 shifts from these potentials. A scaling law for the behavior of viscosity was shown to exist where the quantum calculation goes into the classical value in the limit that h → 0. This scaling law parallels a similar result in which Fraunhoher diffraction increases the hard sphere geometric cross section by a factor of two. The unitary limit for pure neutron matter, when correct for effective range factor, was found to hold over a broad range of temperatures. A simple analytic expression was also given for this system away from the unitary limit. Several other features that are unique to the nuclear system such as a tensor force and an inner hard core or strong repulsion are also mentioned. The inner hard core removes a divergence inη as the index of refraction goes to unity. The entropy and in particular the non-ideal gas interaction entropy was further developed beyond the initial results of Ref. [28,29]. Using results for the viscosity and entropy density, the ratio of the viscosity to entropy density was analyzed both in the unitary limit and away from the unitary limit. The importance of the isothermal compressibility in such studies was pointed out. Theη / s ratio was developed in two variables, the thermodynamic fugacity variable z and a variable y which involves the ratio of either the scattering length or interaction radius to interparticle distance. The minimum in theη / s ratio in the unitary limit (approximately10 times the string theory result of (1/ 4π )h / kB ) was shown to occur at high fugacity and suggests that higher order correlations beyond the two particle case are necessary for a more accurate description. Calculation based on a hard sphere Fermi gas where shown to be several times larger than the unitary limit. The need to include higher order terms was also seen in the behavior of the isothermal compressibility which is negative at the minimum inη / s for attractive interactions. Systems with negative values of the isothermal compressibility are mechanically unstable. The relation of these aspects with a van der Waal gas was developed and the properties associated with a liquid-gas phase transition and a critical point were noted. The results from RHIC suggest a much lower η / s ratio and a more perfect liquid than the moderate energy nucleonic case considered here with a much higher η / s ratio even in the unitary limit. Finally, the importance of the viscosity in reducing the flow was illustrated using both a linear transport theory based on a relaxation time approximation to the Boltzmann equation and also using a Fokker Planck equation. The importance of the Reynolds number was stressed to see if the flow is laminar or turbulent.
Appendix A. Properties of a hard sphere gas
2 2 2 Usingtan δl = jl (x) /ηl (x ) , x = kRC and dδl / dx = −1/ x ( jl (x) +ηl (x)) the viscosity and entropy of a hard sphere gas can be calculated. Identical spin ½ fermions can coupled to total spin 0 in even orbital angular momentum states l = 0,2,4 ,... and can couple to total spin 1 in odd l = 1,3,5 ,... states. Spin 0 bosons only interact in even orbital angular momentum states. At low energies, the S − wave interaction dominates and the ω integral is very accurately described by replacing the sum inφ with just the l = 0 term which is 2 1/ 3 2)sin (x /3. A dimensionless variable y = 2π RC /(V / A) is a measure of the interaction distance to particle separation. Theω , including a factor of 2 for identical
27 2 fermions or bosons and a factor1/ gS =¼ for spin ½ fermions coupled to spin 0, is
2 4 2 1 1 ω(ξ) = 2 4ξ 11/ 2 ( ξ (9ξ − 2) + {4 + 5ξ(3ξ − 4)}⋅ D( )) (A.1) gS 3 8ξ ξ
The D(1/ ξ ) is the Dawson F function which has an expansion for large ξ given by the −1/ 2 n n n D(ξ ) =)ξ Σn=0 (−1) 2 /(ξ (2n −1)!! . An expansion in inverse powers of ξ gives 2 2 3 ω(ξ ) ≈ (2 ⋅8 / g S ) ⋅ (1− 4 / 3ξ + 8 / 9ξ − 8 / 21ξ + ...). The long wavelength limit for 2 2 S − waves is obtained from sin x ≈ x givingω(ξ ) = 2 ⋅8 / g S . The Sint =0 since 1/ 2 BC (ξ ) ~ 1/ ξ . Theη / s is then given by the equation
2 η gS h 5 2 1 1 1 = 2π 2 1/ 3 2/ 3 2 7 / 2 . (A.2) s 2 kB 8 y z ω(z / y ) (5 / 2 − ln z ± z /(2 gS ))
Table 1 summarizes properties of hard sphere Fermi and Bose gases in two extreme limits. Between these limits properties of these gases can be obtained from Eq. A.3-A.5. ______TABLE 1. Limiting behaviors for the interaction entropy density, viscosity, ratio of viscosity to entropy density for spin ½ fermions(gS = 2 ) and spin 0 bosons(gS = 1 ) . ______Variable S − wave long wavelength limit Geometric short wavelength limit
S A2 1 4πR3 int 0 − k C V B V 2 2 3
2 5 2π h 1 5 2π h 1 η gS 2 2 128 λT πRC 16 λT πRC
η 2 s gS 5 2π 2 1 2 h 2 64 y 1/ 3 5 z z ( + ln gS − ln z ± 7 / 2 ) kB 2 2 gS
2 5 2π 2 1 gS 2 3 16 y 1/ 3 5 z y z ( + ln gS − ln z ± 7 / 2 − ) 2 2 gS 3 2π
______
28 The general result for η / s as a function(z, y ) which includes all partial waves is
η 5 2π 2 1 1 = h (A.3) 2 1/ 3) s k 8 y ω(ξ ) 2 / 3 2 z s(ξ ) 2 / 3 2 B ξ = z / y ξ = z / y where
∞ 1 1 (g −1)(g +1) ω(ξ ) = 2 ⋅ 4ξ 4 exp(−ξx2 )x7 F (x) + S S F (x)dx (A.4) ∫ 2 2 E 2 O 0 x gS gS
(l +1)(l + 2) 2 (l +1)(l + 2) 2 FE (x) = ∑ sin (δl +2 − δl ) , FO (x) = ∑ sin (δ l+2 − δ l ) l =0,2,... 2l + 3 l=1,3,,... 2l + 3 and
) 5 z 3/ 2 2 d 1/ 2 ) s(ξ ) = + ln gS − ln z ± 7 / 2 − 2 ξ ξ BC (ξ ) (A.5) 2 2 gS dξ
∞ ) 1 2 1 (g −1)(g +1) B (ξ ) = exp(−ξx ) M (x) + S S M (x) C ∫ 2 E 2 O π 0 gS gS
dδ2l dδ 2l+1 M E (x) = ∑(4l +1) , M O (x) = ∑(4l + 3) l =0,1,.. dx l=0,1,.. dx
3 5 7 The P, D, F wave phase shifts for small x areδ1 = −x / 3 + x / 5 − x / 7 + ..., 5 7 7 2l+1 2 δ2 = −x / 45 + x /189 − ... , δ3 = −x /1575 + ... . The δ l = −x /(2l +1)((2l −1)!!) is the leading order term for eachl . The semi-classical limit has h → 0 andξ → 0 . The D − S 5 2 5 2 4 6 contribution toω(ξ ) involves δ0 − δ2 ~ x − x / 45, has sin (x − x / 45) = x − x / 4 + 0x and is missing the lowest order D − wave interference term. Therefore, the 1/ ξ 2 term arises solely from the P − wave. The contribution of each partial wave is also given. It should be noted that the result of Eq.(18) gives an expansion forω in inverse powers of 2 2 2 h since ξ = (λT / RC 2π ) ~ h . The viscosity is connected to this series expansion around the S − wave scattering limit using Eq.(4). Some remarks regarding the entropy are as follows. A spinless Bose gas has only even l − terms. Therefore, the first contribution comes from a D − wave. Letting Sint,2 be the 2 2 D − wave contribution to Sint an expansion for large ξ = λT / 2πRC gives
29 2 2 3 2 A 20π RC RC Sint,2 = − kB 2 . (A.5) V 3 λT
A P − wave interaction for fermions gives an interaction entropy Sint,1 which is
A2 3 3 π eξ (1+ 2ξ) 1 S = −k 23/ 2 (2πR2 )3/ 2 ξ 2 Erfc( ξ − (A.6) int,1 B 2 C V gS π 2 2 ξ π
2 2 2 Using ξ = b / RC = λT / 2πRC and the small x expansion of the phase shifts leads to an associated largeξ lowT expansion of BC = BC,E + BC,O . The BC,E ,BC,O are the even, odd l 7 / 2 parts of BC . To order 1/ ξ :
1 5 25 3 9 1 15 7 B ( (3 )) (A.7) π C = − 1/ 2 + 5/ 2 − 7 / 2 − 3/ 2 − 5/ 2 + 7 / 2 + 2ξ 24ξ 144ξ E 4ξ 8ξ ξ 16 225 0
A comparison between the S-wave hard sphere Fermi gas and a Fermi gas based on all terms as developed in this appendix is shown in Fig. 9 for the case of y = 0. 4
η y 2 / h s kB 0.53 0.52 0.51 0.50 0.49 0.48
1.01.52.02.53.03.54.0 z
FIG. 9. (Color online) Comparison between the S-wave hard sphere Fermi gas and a Fermi gas based on all terms using Eq. (A.3). The results are fory = 0. 4 . The upper curve is the S − wave result. Some differences exist between the two curves, such as the shift to lower fugacity when higher partial waves are included. However the value at the 2 minimum hasn’t changed significantly. The value of η / s = 3h / kB when y η / s = .48h / kB .
______
30 This work is supported by Department of Energy under Grant DE-FG02ER-409DOE and was done in part at Triumf Laboratory in Vancouver, BC. ______
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